Geometry and Topology of Manifolds with Special Holonomy

特殊完整流形的几何与拓扑

• 批准号: 1105663
• 负责人: Sema Salur
• 金额: $ 12.67万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105663&HistoricalAwards=false
• 关键词: Geometry; Topology; Manifolds; Special; Holonomy

Manifolds with special holonomy and calibrated geometry have important relations with symplectic geometry, complex geometry, algebraic geometry, Cartan-Kahler Theory and Seiberg-Witten Theory. In this project, the P.I. plans to continue her work on calibrated submanifolds of Calabi-Yau, G_2 and Spin(7) manifolds. In joint work with S. Akbulut, the P.I. studied complex associative and complex Cayley submanifolds and using the SW theory she showed that the moduli space of these submanifolds is smooth without any obstructions to such deformations. She also introduced the mathematical definitions of mirror Calabi-Yau and G_2 manifolds. The P.I. plans to continue her work on compactification problems of these moduli spaces and to study the mirror dualities in other Calabi-Yau and G_2 manifolds. These will lead to the construction of new counting invariants and provide a better understanding of the mirror symmetry phenomenon. The research topics of this project are motivated by questions in geometry and topology of low dimensional manifolds and mathematical physics. The P.I. believes that this makes calibrated geometry an excellent subject for students who might decide to study math and science, and as a female researcher in this exciting area she feels a particular responsibility to encourage young women to begin and to continue their studies of mathematics. She will continue to encourage both undergraduate and graduate students and to collaborate with them in this research field which is expected to have a long-lasting impact on both mathematics and physics.

具有特殊完整性和标定几何的流形与辛几何、复几何、代数几何、Cartan-Kahler理论和Seiberg-Witten理论有着重要的关系。在这个项目中，P.I.计划继续她在 Calabi-Yau、G_2 和 Spin(7) 流形的校准子流形方面的工作。在与 S. Akbulut 的合作中，P.I.研究了复杂的关联子流形和复杂的凯莱子流形，并使用 SW 理论证明了这些子流形的模空间是光滑的，没有任何对此类变形的阻碍。她还介绍了镜像Calabi-Yau和G_2流形的数学定义。 P.I.计划继续研究这些模空间的紧致化问题，并研究其他 Calabi-Yau 和 G_2 流形中的镜像对偶性。这些将导致新的计数不变量的构造，并提供对镜像对称现象的更好理解。该项目的研究主题源于低维流形的几何和拓扑以及数学物理问题。 P.I.相信这使得校准几何成为可能决定学习数学和科学的学生的绝佳科目，作为这个令人兴奋的领域的女性研究员，她感到有特殊的责任鼓励年轻女性开始并继续数学研究。她将继续鼓励本科生和研究生，并与他们在这一研究领域进行合作，预计这将对数学和物理学产生持久的影响。